Ever since coins were invented, they have been flipped (tossed). They are flicked up into the air by the thumb, spin several times, and land face-up (Heads) or face-down (Tails).
Kids love to flip them for fun. Grown-ups may do it for fun or more serious matters according to the simple rule of “Heads, I win; Tails, you lose”. In othter words,Heads is defined as Success and Tails as Failure. Here are some real examples.
Gambling
Starting a football game. The winning team gets to choose whether to kick or receive the ball.
The Wright brothers chose who would be the first to fly a heavier-than-air powered flying machine. Wilbur won the toss but crashed on take-off. Orville then tried and succeeded.
Choosing the name of a city in Oregon. Bostton and Portland were candidates. We know who won.
Deciding who will be the first author on a scientific publication
They decided to flip 1 coin 10 times. Mary guessed that the outcome would be more tails than heads, while Sam thought there would be 5 of each possibility. The result of the experiment was 6 heads and 4 tails.
How can they know if their guesses (intuitions) are right? If they are too young to have learned about the mathematical formulas that provide answers, they can interpret the numbers on a chart called a “binomial table”. Binomial means two names. In our case it includes heads and tails or successes and failures.
With a little guidance from someone who knows how to read the chart, they can see thaat Mary’s prediction of 6 tails in 10 trials has a probability of 0.2051, about 1 out of 5. The outcome they actually got was 4 tails. To their surprise, this result has the same probabilty of 0.2051.
Sam’s prediction of 5 Heads out of 10 flips has a probability of 0.2461, or about 1 in 4 trials.
This experiment demonstrates that given random, independent events, outcomes with the highest probability don’t always win the game. This is why people keep buying lottery tickets. They know that the chance of winning is far less than the chance of getting hit by lightning: 1/750,000 = 0.000001. But they also know, incredible as it may seem, that there are, indeed, lucky people who really do hit the jackpot. Long, long ago the Greeks believed that the gods were responsible for random outcomes. Today, even among the well-educated, many try to assist fate with good luck charms. Popular ones include a 4-leaf clover, rabbit’s foot, horseshoe, fuzzy dice, ladybugs, the number 7, and other gris-gris.
Her prediction of more Tails than Heads, is equivalent to fewer Heads than Tails, or at most 4 Heads (0,1,2,3 or 4). Or we could say at least 6 Tails (6,7,8,9 or 10). Since we now know about the symmetry of the distribution of the results, the total probability for this outcome is the same for either interpretaton. The total probability for all possibilities from 0-10 Heads equals 1.0, equals 100 %.
Reading from the chart we find the sum of the probabilites of relevant outcomes:
.0010+.0098+.0439+.1172+.2051 = .3770
## Probabillity of more Tails than Heads = 0.377
## Probabillity of fewer Heads than Tails = 0.377
Law of Large Numbers
Monte-Carlo Simulations
Normal (Gaussian) Distribution
Measures of Variability
Measures of Central Tendency
Central Limit Theorm
And More!